1

Fast predictions of variance images for fan-beamtransmission tomography with quadratic

regularizationYingying Zhang-O’Connor, Student member, IEEE, Jeffrey A. Fessler, Fellow, IEEE

yzz,fessler@eecs.umich.edu

September 5, 2006Abstract— Accurate predictions of image variances can be

useful for reconstruction algorithm analysis and for the designof regularization methods. Computing the predicted variance atevery pixel using matrix-based approximations [1] is impractical.Even most recently adopted methods that are based on localdiscrete Fourier approximations are impractical since they wouldrequire a forward and backprojection and two FFT calculationsfor every pixel, particularly for shift-variant systems like fan-beam tomography. This paper describes new “analytical” ap-proaches to predicting the approximate variance maps of 2Dimages that are reconstructed by penalized-likelihood estimationwith quadratic regularization in fan-beam geometries. The sim-plest of the proposed analytical approaches requires computationequivalent to one backprojection and some summations, so it iscomputationally practical even for the data sizes in X-ray CT.Simulation results show that it gives accurate predictions of thevariance maps. The parallel-beam geometry is a simple specialcase of the fan-beam analysis. The analysis is also applicable to2D PET.

Index Terms— variance approximation, local discrete Fourieranalysis, fan-beam tomography, penalized-likelihood image re-construction.

I. INTRODUCTION

STATISTICAL methods have obtained increasing attentionin tomographic image reconstruction due to improved

noise and resolution properties. These methods are usuallynonlinear and shift-variant. To analyze the statistical character-istics of the reconstructed images, one would like to be able topredict the variances and covariances of estimated pixel values.The variance information provides an uncertainty measure ofthe reconstructed image and may aid regularization parameterselection.

The existing noise analysis methods can be divided into twocategories: iteration based and estimator based. The iteration-based variance predictions are studied in [2], [3] as a functionof the iteration number for the maximum-likelihood expec-tation maximization algorithm based on the “stopping rule”to terminate the iterations before convergence. The estimator-based variance predictions are independent of the particularalgorithm and iterations, [1], [4], [5]. Our proposed methodfalls in the estimator-based category. We will give a briefoverview on the existing estimator-based methods and ourproposed method.

This work is supported in part by NIH/NCI grant P01 CA87634 and GEMedical Systems.

EECS Dept., The University of Michigan, Ann Arbor, MI 48109-2122

The estimator-based analysis for the mean and varianceproposed in [1] uses the partial derivatives of the cost functionand Taylor approximations. The approximations are matrixform and give accurate results. However, the predictionsinvolve the inversion of the Hessian matrices and thereforeare computationally expensive. Based on this work, a greatdeal of effort has been given to simplify these matrix meth-ods [4], [5]. All these methods, that we refer to the DFTapproximations, are based on a factorization of the systemmatrix and circulant approximations to the Hessian matricesto precompute and store a great portion of the calculations. Thefactorization of the system matrix into geometric and object-dependent portions is specially useful for the shift-varyingimaging systems. However, these DFT approximations stillrequire in precomputation one forward and backprojection andtwo FFT calculations, one for likelihood Hessian and one forpenalty Hessian, for each location of interest. Moreover, theexpressions are still in matrix form and provide little directinsight into the noise properties.

Our proposed approximations in this paper are still basedon the results given in [1] but turn to a very differentanalysis approach. Instead of working in the discrete space,we use the discrete space Fourier transform (DSFT) andParseval’s theorem to bridge from the discrete space to thecontinuous space. Using local shift-invariance approximationsand local Fourier analysis, we derive “analytical” closed-formexpressions for the local impulse response and local frequencyresponse of the Gram operator and the regularization operator.The final approximations eliminate the need of FFTs forvariance predictions, greatly reducing computation for caseswhere the variance is to be predicted at numerous pixellocations. Furthermore, these approximations provide insightinto the resolution and noise properties of the reconstructedimages.

Because our analysis is built on the previous work in [1], webriefly repeat its main results here. The goal of transmissionimage reconstruction is to estimate an attenuation image µ[~n]from projection data Y , where ~n is a vector denoting the 2Dimage pixel location. We focus here on penalized-likelihoodestimators obtained by minimizing a cost function as follows:

µ = arg minµ

Φ(µ,Y ),

where µ = (µ[~n1], . . . , µ[~np]) ∈ Rp (p-dimensional real

space). The cost function includes a negative log-likelihood

2

PSfrag replacementsnm

(0, 0) (N − 1, 0)

(0,M − 1) (N − 1,M − 1)

~n1 ~n2

~n3 ~n4 ~n5 ~n6

~n7 ~n8 ~n9 ~n10

~n11 ~n12

Fig. 1. A N × M lattice with approximately circular FOV. Only the pixelswith indices are estimated. In this example, p = |S| = 12.

term and a regularization term:

Φ(µ,Y ) = −L(µ,Y ) + αR(µ). (1)

As a concrete example, for transmission tomography under thePoisson noise model, the log-likelihood is

L(µ,Y ) =∑

i

Yi log(

Yi(µ))

−Yi(µ). (2)

For mono-energetic transmission scans, the measurementmeans are modeled by

Yi(µ) = bi e−[Aµ]i + ri, (3)

where A is the system matrix, bi denotes the blank scan, andri denotes the additive contribution of scatter to the ith ray.

We focus on regularization terms of the following form:

R(µ) =

4∑

l=1

Rl(µ) (4)

Rl(µ) =∑

~n,~n−~ml∈S

rl[~n]1

2(µ[~n]− µ[~n− ~ml])

2, (5)

where S , ~nj : j = 1, . . . , p denotes the sub-set of the N × M lattice that is estimated and ~ml ∈(1, 0), (0, 1), (1, 1), (−1, 1). The roughness penalty (4)involves the horizontal, vertical, and diagonal neighbors andallows for the possibility of using regularization coefficientsrl[~n] that vary both with spatial location and direction [6],[7]. In general 1 ≤ p ≤ NM and p < NM because thephysical field of view (FOV) is a subset of the lattice, seeFig. 1.

The goal of this work is to approximate the covariancematrix Covµ efficiently yet accurately, motivated by theproblem of designing the regularizer R(µ). The proposedprediction methods can be generalized to other log-likelihoodterms including 2D emission tomography by modifying W in(??) below.

The following approximation to the p×p covariance matrixof µ was derived in [1]:

K = (A′WA + αP)−1A′WA(A′WA + αP)−1, (6)

where P is the Hessian matrix of the roughness penalty. Fortransmission tomography with the models (1) and (2), W =

diag

Yi

. In practice Yi is unknown, so we plug in Yi as anapproximation [8]. The covariance between pixels µ[~nk] andµ[~nj ] can be approximated using (6) as follows:

Covµ[~nj ], µ[~nk] ≈ e′jKek, (7)

where ej denotes the jth unit column vector of length p.The matrix method described in (6) and (7) has been used

in various applications [5], [9]. Simulation and experimentalresults have confirmed the accuracy of this covariance approx-imation in image regions where the non-negativity constraintis usually inactive. However, evaluating (7) is relatively expen-sive. In this paper, we introduce “continuous space analysis”and use “local stationarity” to develop fast approximations forthe variance and covariance of the reconstructed image µ[~n].

The paper is organized as follows. Section II briefly reviewsthe matrix method and the local shift-invariance approxima-tions. Section III proposes the general analytical approachfor the variance approximation. Section IV and V apply thismethod to fan-beam geometry and quadratic regularization.Section VI and VII analyze the single integral approach usedand give simulation results for two types of quadratic regular-ization, including a comparison of the predicted, DFT-basedand empirical standard deviation images. Finally, discussionand conclusions are in Section VIII.

II. LOCAL SHIFT-INVARIANCE APPROXIMATIONS

The matrix method described in (6) and (7) is very expen-sive to compute, even for the variance at a single pixel. Toaccelerate computation, local shift-invariance approximationsare usually used in practice, e.g., [4], [5], [9]–[11].

Let M denote one of the p × p matrices in (6), such asA′WA or P, or inverses or sums thereof. Then a matrix-vector operation y = Mx can be expressed equivalently as

y[~n] = δS [~n]∑

~n′∈S

h(~n, ~n′) x[~n′]

= δS [~n]∑

~n′

h(~n, ~n′) x[~n′]δS [~n′], (8)

where δS [~n] is an indicator function of ~n defined as follows:

δS [~n] ,

1, ~n ∈ S0, otherwise . (9)

In other words, the elements of M correspond to Mkj =h(~nk, ~nj) .

Near a given location ~n0 of interest, we define a localimpulse response of M as follows1:

h0(~m) , h(~n0 + λ~m,~n0 − (1− λ) ~m)

δS [~n0 + λ~m]δS [~n0 − (1− λ)~m], (10)

where ~m ∈ Z2, Z denotes the set of integers. Usually we

choose λ = 1. However, sometimes we can approximate heven for non-integer arguments, in which case λ = 1/2 mayalso be useful [12, p. 870].

We say that h(~n, ~n′) is locally shift invariant near ~n0

if h(~n, ~n′) ≈ h0(~n− ~n′) for ~n and ~n′ close to ~n0. The

1Throughout the paper we use the subscript “0” to indicate dependence ona given pixel location ~n0.

3

approximation should be accurate provided ~n and ~n′ are“sufficiently close” to ~n0 relative to the width of h0. Thus,if the operator M is approximately locally shift invariant near~n0, then we can approximate the superposition sum (8) by(almost) a convolution sum:

y[~n] ≈ δS [~n]∑

~n′

h0(~n− ~n′) x[~n′]δS [~n′], (11)

or equivalently y ≈ M0x, where the p × p matrix M0 isdefined by [M0]kj = h0(~nk − ~nj) . The expression (11) isalmost a convolution sum, except for the “edge conditions” ofthe indicator functions. If the point ~n0 is not “too close” tothe boundaries of the support mask S, then we may able todisregard the indicator functions and treat the expression as aconvolution.

Let T be the NM × p matrix such that

T1+n+mN,j =

1, ~nj = (n,m)0, otherwise ,

for n = 0, . . . N − 1 and m = 0, . . . M − 1. The purposeof T is to embed the p elements of µ (as shown in Fig. 1)back to the 2D N ×M lattice. Then M0 = T

′M0T, where(M0)~n,~n′ = h0(~n− ~n′) is an NM×NM matrix that is blockToeplitz with Toeplitz blocks (BTTB). Thus we can make acirculant approximation to M0, [13]. Such approximations areoften reasonably accurate except near the edges of the FOV,where the differences between “Toeplitz” and “circulant” endconditions are largest. The local impulse response (10) and thecorresponding circulant approximation are two key tools foranalysis.

III. THE ANALYTICAL VARIANCE PREDICTION

In the spirit of the local shift-invariance approximationspresented in Section II, we approximate the covariance matrixin (6) near a given location ~n0 by

K ≈ K0 , T′K0T

K0 , (F0 + αP0)−1F0(F0 + αP0)

−1,

where F0 and P0 are the NM ×NM BTTB approximationscorresponding to A′WA and P, respectively. Then we ap-proximate the covariance between pixels µ[~n] and µ[~n′] in (7)by the following inner product:

Covµ[~n], µ[~n′] ≈ 〈K0e~n′ , e~n〉, (12)

where e~n is ~nth unit vector of length NM .Two useful approximations to (12) follow from Parseval’s

theorem. One option is to interpret the arguments in (11) witha suitable modulo N or M . In this case, the inner productdefined in (12) is in the form of circulant convolution and canbe approximated by FFTs:

Covµ[~n], µ[~n′] ≈ 1

NM

~N−1∑

~k=~0

Pd0[~k] ei~ω~k·(~n−~n′) , (13)

for ~n, ~n′ ≈ ~n0, where ~N = (N,M), ~ω~k =(2πk1/N, 2πk2/M) and

Pd0[~k] ,Γ0[~k]

(Γ0[~k] + αΩ0[~k])2,

with

F0 ≈ QΓ0Q′

P0 ≈ QΩ0Q′,

where Q is the 2D (N,M )-point orthonormal DFT matrix.The diagonal matrices Γ0 and Ω0 have diagonal elementsΓ0[~k] and Ω0[~k] that are the 2D DFT coefficients of the localimpulse response of A′WA and P near ~n0, respectively. ThisDFT/FFT approximation has been used in [4], [14], [15] topredict variance at a single pixel:

Varµ[~n] ≈ 〈K0e~n, e~n〉

≈ 1

NM

~N−1∑

~k=0

Γ0[~k]

(Γ0[~k] + αΩ0[~k])2. (14)

Generally, evaluating this expression for a single pixel requiresa forward and backprojection and two FFTs. Computation ofthis DFT approximation is still expensive for realistic imagesizes when the variance must be computed for many or allpixels, particularly for shift-variant systems like fan-beamtomography.

An alternative option is to consider µ[~n] to be defined overall of Z

2 (two-dimensional integer space), in which case (12)is in the form of ordinary convolution that can be expressedusing the discrete-space Fourier transform (DSFT) as follows:

Covµ[~n], µ[~n′] ≈∫ π

−π

∫ π

−π

Pd0(~ω) ei~ω·(~n−~n′) d~ω

(2π)2, (15)

where Pd0(~ω) is the local spectrum of K0, given as follows:

Pd0(~ω) ,Hd0(~ω)

[Hd0(~ω) + αRd0(~ω)]2, (16)

where Hd0(~ω) is the local frequency response of the Grammatrix A′WA and Rd0(~ω) is the local frequency response ofP near ~n0. To our knowledge, this paper is the first to use(15) to develop analytical variance approximations as a fasteralternative to the DFT approach (14).

For regularizer design, the standard deviation map of thereconstructed image is one quantity of interest, and our numer-ical investigation will focus on variance prediction. However,the methodology applies readily to approximate covariances.

Using the DSFT approximation (15), we approximate thevariance at pixel ~n0 as follows:

Varµ[~n0] ≈∫ π

−π

∫ π

−π

Pd0(~ω)d~ω

(2π)2. (17)

Let ∆ denote the sample spacing in the reconstructedimage. By making the change of variable, ~ω = (2πρ∆)~eΦ

where ~eΦ , (cos Φ, sin Φ), we rewrite (17) in terms of polarfrequency coordinates (ρ,Φ) as follows:

Varµ[~n0] ≈ ∆2

∫ 2π

0

∫ ρmax

0

P0(ρ,Φ)ρdρ dΦ, (18)

where ρmax = 12∆ , and we define

P0(ρ,Φ) , Pd0(2πρ∆~eΦ) =H0(ρ,Φ)

[H0(ρ,Φ) + αR0(ρ,Φ)]2.

(19)

4

X−raysource

detectorPSfrag replacements

γ

ϕ

ϕ

Ds0β

rs

Dsd

Fig. 2. Angular coordinates in fan-beam geometry.

We defined H0 and R0 similarly in terms of Hd0 and Rd0. Thevariance prediction (18) applies to any 2D geometry. The nextsection specializes (18) by finding analytical approximationsto the local frequency response H0(ρ,Φ) for the fan-beamgeometry.

IV. FAN-BEAM GEOMETRY

The following analysis is focused on equiangular fan-beamtransmission tomography with an arc detector. However, themethod generalizes readily to flat detectors, i.e., equidistantsampling and to parallel-beam geometries. As illustrated inFig. 2, fan-beam rays are indexed by coordinates (s, β), whereβ is the angle of the source relative to the y axis, and s is thearc length along the detector. For the case where the detectorfocal point is at the source position, γ(s) = s/Dsd, where γ isthe angle of the ray relative to the source and Dsd is the sourceto detector distance. The relation between parallel-beam andfan-beam coordinates is [16]:

r(s) = Ds0 sin γ(s) (20)ϕ(s, β) = β + γ(s), (21)

where Ds0 is the source-to-rotation center distance.

A. Local Impulse ResponseTo predict variance images in fan-beam transmission tomog-

raphy using (18), we need to determine the local frequencyresponse H0(ρ,Φ), or equivalently Hd0(~ω). We first find thelocal impulse response.

Consider the 2D object model based on a common basisfunction χ(~x) superimposed on a N ×M Cartesian grid asfollows:

µ(~x) =∑

~n∈S

µ[~n]χ

(

~x− ~xc[~n]

∆

)

, (22)

where ~x ∈ R2 denotes the 2D coordinates of the continuous

image space, and ~xc[~n] denotes the center of the basis function.Typically

~xc[~n] = (~n− ~w~x)∆, ~n ∈ S~w~x = ( ~N −~1)/2 + ~c~x,

where ~N = (N,M) and the user-selectable parameter ~c~x

denotes an optional spatial offset for the image center.For simplicity, we assume here that the detector blur b(s) is

locally shift invariant, independent of source position β, andacts only along the s coordinate. Then we model the meanprojections as follows:

yβ [sk] =

∫

b(sk − s′) pϕ(s′,β)(r(s′)) ds′ (23)

for sk = (k − wS)∆S and k = 1, . . . , ns, where ∆S is thesample spacing in s, wS is defined akin to ~w~x, and pϕ(r) isthe 2D Radon transform of µ(~x):

pϕ(r) =

∫

µ(r cos ϕ− ` sin ϕ, r sinϕ + ` cos ϕ) d` .

Substituting the basis expansion model in (22) for the objectinto the measurement model (23) and simplifying leads to thelinear model

yβ [sk] =∑

~n∈S

a(sk, β;~n)µ[~n],

where the fan-beam system matrix elements are samples ofthe following fan-beam projection of a single basis functioncentered at ~xc[~n]:

a(s, β;~n)=

∫

b(s− s′) ∆ g

(

r(s′)− rϕ(s′,β)[~n]

∆, ϕ(s′, β)

)

ds′,

(24)where g(·, ϕ) is the Radon transform of χ(~x) at angle ϕ and

rϕ[~n] , ~xc[~n] · ~eϕ,

with ~eϕ , (cosϕ, sinϕ).Then the elements of the Gram matrix are given exactly by

hd[~n;~n′] =

[A′WA]jj′ , ~n = ~nj ∈ S, ~n′ = ~nj′ ∈ S0, otherwise

= hd[~n;~n′]η(~xc[~n])η(~xc[~n′]) (25)

where η(~xc[~n]) , 1~n∈S,

hd[~n;~n′] ,

nA∑

l=1

ns∑

k=1

w(sk, βl) a(sk, βl;~n)a(sk, βl;~n′) (26)

and w(s, β) denotes the weighting associated with W andnA denotes the number of samples of the source position β.To simplify (25), we first use an integral to approximate thesummation in (26) as follows:

hd[~n;~n′]≈ 1

∆β

1

∆S

∫ 2π

0

∫ ∞

−∞

w(s, β) a(s, β;~n)a(s, β;~n′)ds dβ,

(27)where ∆β is the source angular sampling interval. Notice thathd[~n;~n′] in (27) is not shift invariant.

We develop locally shift-invariant approximations tohd[~n;~n′] in (27) by reparameterizing variables s, β usinganalogs of fan-to-parallel beam rebinning. The following lo-cally shift-invariant approximation to hd[~n;~n′] is derived indetail in Appendix I:

hd[~n;~n′] ≈ 1

∆β

1

∆S

∫ 2π

0

w0(ϕ)h0(∆(~n− ~n′) · ~eϕ, ϕ) dϕ, (28)

5

where the following 1D autocorrelation is with respect to r:

h0(r, ϕ) , a0(r, ϕ) ? a0(r, ϕ),

and a0(r, ϕ) is a locally parallel-beam version of the systemmodel defined in (51) (see Appendix I). The angle-dependentweighting w0(ϕ) is associated with pixel ~n0, accounting forthe position-dependent magnification as follows:

w0(ϕ) , |m0(ϕ)|w(s(r0(ϕ)), β(r0(ϕ), ϕ)) (29)r0(ϕ) , rϕ[~n0]

m0(ϕ) ,∂

∂rs(r)

∣

∣

∣

∣

r=r0(ϕ)

=Dsd/Ds0

√

1− (r0(ϕ)/Ds0)2, (30)

where s(r) and β(r, ϕ) are the inverse of (20) and (21). Theshape of the local impulse response (28) is a modification of1/r (cf [17]) with statistically modulated angular weighting.The key property of (28) is that it is locally shift invariant,except for edge effects. This approximation should be reason-ably accurate provided that ~n and ~n′ are “sufficiently close”to ~n0, the coordinates of the pixel of interest.

B. Local frequency responseHaving found the local impulse response approximation

(28), the next step is to find the local frequency response.This requires consideration of the edge effects in (25).

The following local frequency response near a point ~n0 isderived in detail in Appendix II:

H0(ρ,Φ) ≈ 1

∆β∆S∆2

∫ 2π

0

w0(ϕ)Sϕ(ρ,Φ) dϕ, (31)

where the following function captures both detector responseeffects and edge effects:

Sϕ(ρ,Φ) = |A0(ρ cos(Φ− ϕ), ϕ)|2

·d0(ϕ) sinc2(d0(ϕ)ρ sin(Φ− ϕ)), (32)

d0(ϕ) denotes the length of the chord through ~n0 through theFOV at angle (ϕ+π/2), and A0(ν, ϕ) is the 1D FT of a0(r, ϕ)with respect to r.

C. Further approximations of local frequency responseThe local frequency response of the Gram operator in (31)

is very accurate. However, direct implementation of (31) isstill computationally demanding. We present here two typesof further approximations to simplify (31).

1) Type I non-separable form: As d0(ϕ) → ∞, one canshow that for large |ρ|,

d0(ϕ) sinc2(d0(ϕ)ρ sin(Φ− ϕ))→ δ(ρ sin(Φ− ϕ)) .

Therefore the sinc2 term is sharply peaked near Φ = ϕand Φ = ϕ ± π, so we consider the further simplifyingapproximation∫ 2π

0

w0(ϕ)Sϕ(ρ,Φ) dϕ ≈ w0(Φ) |A0(ρ,Φ)|2 G0(ρ,Φ),

(33)where

G0(ρ,Φ) =

∫ 2π

0

d0(ϕ) sinc2(d0(ϕ)ρ sin(Φ− ϕ)) dϕ . (34)

0.005 0.01 0.015 0.02 0.0250

1

2

3

4

5

6

7

8

9

10 x 105

ρ (cycles / mm)

loca

l fre

quen

cy re

spon

se H

(ρ,0

)

Type I and II local frequency responses in unweighted case

H01(ρ, 0)H02(ρ, 0)

Fig. 3. Type I and Type II local frequency responses H01(ρ, 0) andH02(ρ, 0) for ~n0 at image center in unweighted case: w(s, β) = 1.H01(0, 0) and H02(0, 0) are not shown because the value of H02(0, 0)blows up.

Substituting into (31) leads to the “Type I” approximation:

H0(ρ,Φ)≈H01(ρ,Φ),w0(Φ)

∆β∆S∆2|A0(ρ,Φ)|2 G0(ρ,Φ).

(35)Although H01(ρ,Φ) is not separable, we can precomputew0(Φ) and tabulate G0(ρ,Φ) once for all pixels for coarselysampled Φ. Accurately computing G0(ρ,Φ) is crucial, there-fore finely sampled ϕ is necessary in (33).

2) Type II separable form: We can simplify further by usingthe sifting property of the Dirac impulse:

∫ 2π

0

w0(ϕ)Sϕ(ρ,Φ) dϕ ≈ 2

|ρ| w0(Φ) |A0(ρ,Φ)|2 .

Because typically A0(ν, ϕ) varies slowly, we also considerthe following further approximation:

A0(ν, ϕ) ≈ A0(0, ϕ).

Combining all the above approximations yields the followingseparable approximation to the local frequency response:

H0(ρ,Φ) ≈ H02(ρ,Φ) ,2 |A0(0, ϕ)|2∆β∆S∆2

w0(Φ)

|ρ| . (36)

This “Type II” separable form agrees with the familiar FTof 1

r . Figure 3 shows the profiles of two types of localfrequency responses for ~n0 at image center in unweightedcase. We can see that two profiles agrees with each otherclosely. The discrepancy is mainly at low frequencies for boththe unweighted and weighted cases. The discrepancy is moreobvious when ~n0 is off image center.

V. QUADRATIC REGULARIZATION: R0(ρ,Φ)

To evaluate the variance using (18) and (19), we alsoneed the local frequency response of quadratic regularization,R0(ρ,Φ), [7], [8], [18], [19].

Practical regularization methods are based on the differencesbetween neighboring pixel values. For a discrete-space 2Dobject µ[~n], a typical quadratic roughness penalty is given

6

in (4) and (5) for 1st-order differences. The rl[~n] values arepossibly space variant. For the purpose of local frequencyresponse analysis, we examine the characteristics of R(µ) neara pixel ~n0 of interest, so we define rl,0 , rl[~n0] assuming rl[~n]values vary smoothly. Then, the quadratic roughness penaltynear a pixel ~n0 has the following form:

R(µ) =∑

~n

L∑

l=1

rl,01

2

(

(cl ∗∗ µ)[~n])2

.

The rl,0 values are design parameters that affect the direc-tionality of the regularization and hence the shape of the PSF.Each cl[~n] is a (typically) high-pass filter. For a first-orderdifference:

cl[~n] = ξl (δ2[~n]− δ2[~n− ~ml]) ,

or for a 2nd-order difference:

cl[~n] = ξl(δ2[~n]− δ2[~n− ~ml]) ∗∗ ξl(δ2[~n]− δ2[~n− ~ml]),

where ξl = ‖~ml‖−υ/2, ~ml = (nl,ml) denotes the spatialoffsets to the neighboring pixels, and υ is the power of weightsfor diagonal neighbors that can be chosen by the user. Forexample, common practice chooses υ = 1 [20], [21].

Applying Parseval’s theorem, we can rewrite R(µ) asfollows:

R(µ) =

L∑

l=1

∫ π

−π

∫ π

−π

1

2rl,0 |Cl(~ω)U(~ω)|2 d~ω

(2π)2, (37)

where µ[~n]FT←→ U(~ω) and the DSFT of a Λ-order (where

Λ ∈ N) difference has the following magnitude:

|Cl(~ω)| = ξΛl

∣

∣

∣1− e−ı(~ω·~ml)

∣

∣

∣

Λ

= ξΛl 2Λ sinΛ

(

1

2(~ω · ~ml)

)

.

In the polar coordinates of (19):

|Cl(ρ,Φ)|2= |Cl(2πρ∆~eΦ)|2= ξ2Λl 4Λ sin2Λ(π∆ρ~eΦ · ~ml) .

(38)Thus, the Type I local frequency response for the regularizationoperator is

R0(ρ,Φ) = R01(ρ,Φ) =L∑

l=1

rl,0 |Cl(ρ,Φ)|2

=

L∑

l=1

rl,0ξ2Λl 4Λ sin2Λ(π∆ρ~eΦ · ~ml) . (39)

Applying the approximation sin(x) ≈ x to (38) yields:

|Cl(ρ,Φ)|2 ≈ ξ2Λl (~ml · ~eΦ)2Λ(2π∆ρ)2Λ

= (2πρ∆)2Λξ(1−2/υ)2Λl cos2Λ(Φ− ϕl),

where the angle between the lth neighbors is

ϕl , tan−1 ml

nl.

With this simplification, the Type II local frequency responseof the regularizer is approximately separable in (ρ, Φ):

R0(ρ,Φ) ≈ R02(ρ,Φ) = (2πρ∆)2ΛR0(Φ), (40)

where

R0(Φ) ,

L∑

l=1

ξ(1−2/υ)2Λl rl,0 cos2Λ(Φ− ϕl) .

This separable form agrees with the familiar FT of thedifferentiation operation.

VI. VARIANCE PREDICTION IMPLEMENTATION

Having obtained the approximations to H0(ρ,Φ), the localfrequency response of the Gram operator given in (35) and(36), and to R0(ρ,Φ), the local frequency response of theregularizer given in (39) and (40), we can discretize theintegral (18) again to compute the variance image. There aretwo variance prediction expressions for fan-beam transmissiontomography based on the Type I H01(ρ,Φ) given in (35) andR01(ρ,Φ) given in (39), and the Type II H02(ρ,Φ) given in(36) and R02(ρ,Φ) given in (40).

A. Double integral approachThe variance prediction using H01(ρ,Φ) in (35) and

R01(ρ,Φ) in (39) involves a double integral and can beimplemented by a double summation. We call this predictionthe double integral (DI) approach. The location-dependentweighting function w0(Φ) can be precomputed, with the com-putation equivalent to one back-projection. We can coarselysample Φ because P0(ρ,Φ) is fairly smooth along Φ.

B. Single integral approachThe separability properties of H02(ρ,Φ) in (36) and

R02(ρ,Φ) in (40) enable us to carry out the inner integralover ρ analytically. Therefore the double-integral in (18) isreduced to one single integral over Φ. Substituting (36) and(40) into (18) yields the remarkably simple expression:

Varµ[~n]≈∫ 2π

0

ζ/3

2 |A0(0,Φ)|2 w0(Φ) + α4π2ζR0(Φ)dΦ,

(41)where ζ = ρ3

max∆β∆S∆4 is a constant. We call this prediction

the single integral (SI) approach. We evaluate this integralusing a finite summation, with w0(Φ) and R0(Φ) precom-puted. Therefore, computing (41) is equivalent to one back-projection.

C. Implementation of the single integral predictionWe found empirically that the SI approach (41) gave pre-

dictions that could be improved by including a single globalscale factor, presumably because the SI approach (41) usesmany approximations to achieve its simple form. In particular,we found that the SI method under estimates the variance,presumably because the “Fisher information” implied by TypeII local frequency response in (36) is too large for low spatialfrequencies. To determine the scale factor, we assumed thatthe DFT-based approach and the analytical approach shouldproduce equivalent results at the image center. Specifically,we used the predicted variance for unweighted least squares

7

estimator with standard quadratic penalty (QPULS) for unitvariance data.

For QPULS estimator for unit variance data, the statisticalweighting, w(s, β) becomes 1. Consider the standard quadraticpenalty with first-order (Λ = 1) differences and second-orderneighborhood (L = 4), for which ϕ1,2,3,4 = 0, π/2, π/4, and−π/4 and rl,0 = (1, 1, 1, 1). We used υ = 1 both in calibrationand reconstruction as is the common practice in quadraticregularization. For these choices, the Type II R02(ρ,Φ) in (40)becomes independent of Φ:

R0(Φ) =

L∑

l=1

‖~ml‖ = 1 +√

2. (42)

Finally, to determine the scale factor, we computed theratio of the variance predicted by the DFT approach over thatpredicted by (41). For the parameters used in our simulations,this factor was (1.13)2. This value would need to recomputedfor different system geometries or regularization parameters,but is otherwise patient independent.

VII. SIMULATION RESULTS

To evaluate the performance of the proposed methods,we implemented the variance predictions for fan-beam to-mographic images reconstructed by quadratically penalizedlikelihood algorithm. We simulated 1000 realizations of fan-beam transmission scans using a 256×256 Zubal phantom [22]and a blank scan of 106 counts/detector. The correspondingsinogram size was 444 samples in s, spaced by ∆S ≈ 2mm and 492 source positions over 360

. We simulated thegeometry of the GE LightSpeed Pro CT scanner with thesource-to-detector distance around 949 mm, the isocenter-to-detector distance 408 mm and ∆ = 500/256 mm.

An ellipse support was used for S, with p = 43892. Forsimplicity, in (34) we used the width of the central profilethrough the FOV:

d0(ϕ) ≈ d(ϕ) ,2z1z2

√

z21 sin2ϕ+z2

2 cos2ϕ, (43)

where z1 = 244.1 mm and z2 = 220.7 mm are the semi-major and semi-minor axes of the ellipse. This approximationis exact when ~n0 is at the ellipse center. The approximationto d(ϕ) becomes less accurate as ~xc[~n0] approaches the edgeof the ellipse support.

For simplicity, we model the detector response2 in (23) bya shift-invariant rectangle of width ∆S:

b(s) =1

∆S

rect

(

s

∆S

)

.

In the case of a square pixel basis χ(~x) = rect2(~x)3, we havefrom (51) (see Appendix I)

A0(ν, ϕ) = sinc

(

∆Sν

m0(ϕ)

)

∆2 sinc(ν∆cosϕ) sinc(ν∆sinϕ),

(44)2A more accurate model could include detector deadspace and crosstalk

effects.3rect2(~x) , rect(x)rect(y) is a 2D square function.

which we substitute into (32). In our simulation, we make thefollowing simplification:

m0(ϕ) ≈ mc(ϕ) = mc = Dsd/Ds0,

where mc is the value of m0(ϕ) at the image center.We chose the regularization parameter α = 211 to give

FWHM = 1.72 pixels, i.e., 3.4 mm, at the center of the image.For each realization, µ was reconstructed using 70 iterationsof the convergent incremental optimization transfer algorithm(PL-IOT) [23] with 41 subsets and no nonnegativity constraint.The initial images were the filtered back-projection (FBP)images with equivalent spatial resolution, obtained by post-filtering ramp-filtered FBP images with the designed PSF. Wecomputed the sample standard deviation pixel by pixel withinthe finite support S used in reconstruction. All images andprofiles are shown in Hounsfield unit (HU).

Two prediction approaches are investigated here: the DI ap-proximation (18) with Type I H01(ρ,Φ) in (35) and R01(ρ,Φ)in (39), and the SI approximation (41) with R0(Φ) in (40). Theformer formula was derived with fewer approximations whilethe latter formula involves more approximations. The accuracyand computation time are compared below.

We considered two types of regularization: standard andcertainty-based [24]. In both cases, we implemented (39) and(40) for regularization with first-order (Λ = 1) differencesand second-order neighborhood (L = 4). In both cases, thestandard deviation images predicted by the DI approach aredisplayed while both DI and SI predictions are compared inthe profile plots.

A. Standard Quadratic Penalty FunctionWe first considered a standard quadratic penalty where

rl,0 = κ2c ,

and κ2c is the value of κ2[~n0] at the image center in (45) below.

This choice assures that the resolution of the two studies ismatched at the image center. R0(Φ) = (1+

√2)κ2

c is a constantfor all pixels.

Figure 4 shows the reconstructed images and empiricalstandard deviation images. The empirical standard deviationimage for FBP is also shown. The average of FBP standarddeviations is around 2.2 HU, approximately 1.8 times higherthan that of PL-IOT, 1.2 HU, illustrating the noise advantageof the statistical reconstruction methods at matched resolution.

Figure 4 also shows the central horizontal and vertical pro-files of the standard deviation maps. The analytical, the FFT-based and the empirical standard deviations agree with oneanother very closely within the object. The largest discrepancywithin the object was about 10% in the left lung for unknownreasons.

B. Certainty-based Quadratic Penalty FunctionWe next investigate a more complicated regularizer that was

designed to achieve nearly uniform spatial resolution [24]. Inthis case, we used space-varying regularizer:

rl,0 = κ2[~n0],

8

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

pixels

std.

dev

. (HU

)

vertical profiles for case1: κ2c

empiricalprediction w/ double−integralprediction w/ single−integralFFT method

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

pixels

std.

dev

. (HU

)

horizontal profiles for case1: κ2c

empiricalprediction w/ double−integralprediction w/ single−integralFFT method

true phantom Mean PL−IOT recon. Mean FBP w/ matched resol.

0

1200

predicted std. dev. PL−IOT empirical FBP empirical (divided by 1.8)

0

2

Fig. 4. Predicted and empirical standard deviation images (in HU) and central profiles for Zabul phantom for PL fan-beam transmission image reconstructionusing the standard quadratic penalty: R0 = (1 +

√2)κ2

c.

where

κ2[~n0] ,1

2π

∫ 2π

0

w(s(r0(ϕ)), β(r0(ϕ), ϕ)) dϕ . (45)

Here, R0(Φ) is still independent of Φ, but varies spatially.Computing the “certainty map” (45) requires a simple back-projection with fan-to-parallel beam rebinning.

Figure 5 shows the reconstructed images, standard deviationimages and central horizontal and vertical profiles. In thiscase, the average of FBP standard deviations is around 2.2HU, approximately 1.8 times higher than that of PL-IOT, 0.7HU. The analytical, the FFT-based and the empirical standarddeviations agree with one another very closely within theobject.

In both cases, the analytical and the FFT-based predictionsare somewhat less accurate near the edge of the finite supportused in image reconstruction. This is probably due to the factthat the “local stationarity” approximation is less accurate inthis area where the statistical weights w(s, β) can vary rapidly.The approximation (43) may also vary rapidly in our study,so it may be possible to improve accuracy near the edges ofthe FOV by using d0(ϕ).

C. Computation Time and AccuracyIn our calculations, we used 123 samples in Φ and 128

samples in ρ in (18) to predict a 256×256 standard deviationimage. Both DI and SI predictions precompute w0(Φ) andG0(ρ,Φ). The precomputation time for w0(Φ) was about 19

9

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

pixels

std.

dev

. (HU

)

horizontal profiles for case2: κ20

empiricalprediction w/ double−integralprediction w/ single−integralFFT method

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

pixels

std.

dev

. (HU

)

vertical profiles for case2: κ20

empiricalprediction w/ double−integralprediction w/ single−integralFFT method

true phantom Mean PL−IOT recon. Mean FBP w/ matched resol.

0

1200

predicted std. dev. PL−IOT empirical FBP empirical (divided by 3)

0

2

Fig. 5. Predicted and empirical standard deviation images (in HU) and central profiles for Zubal phantom for PL fan-beam transmission image reconstructionusing the certainty-based quadratic penalty.

seconds on dual Intel Xeon 3.40GHz CPU. The precomputa-tion time for G0(ρ,Φ) with 492 samples in ϕ was 2.3 seconds.The DI prediction requires no scale factor precomputation andthe computation time was about 135 seconds. The SI predic-tion requires the scale factor precomputation that is (1.13)2 inour case, and the computation time for prediction was about0.6 second. In contrast, the FFT-based prediction needed about374 seconds to compute only one single central profile. Asexpected, the DI prediction is slightly more accurate thanthe SI prediction, particularly near edges. The SI predictionmatches a bit better with the FFT-based prediction becausethe scale factor calibration was based on FFT-predicted values.For the purposes of regularization design or noise exploration,we believe that the very fast SI approach is adequate.

Because we only compute two central profiles of the FFT-based prediction in each case, we compute the normalized rms(NRMS) percent errors only for these two central profiles. Forκ2

c case, the NRMS percent errors for FFT, DI and SI are6.6%, 6.8% and 6.6%; for κ2

0 case, the NRMS percent errorsfor FFT, DI and SI are 6.5%, 6.0% and 8.3%, respectively.

VIII. CONCLUSION AND DISCUSSION

This paper has developed analytical variance approxima-tions for 2D tomography. The double integral (18) with(35) and (39), and the single integral (41) provide fast andaccurate variance predictions for a quadratically penalizedlikelihood estimator in fan-beam tomography. The simplest ofthe proposed approaches (41) requires one backprojection with

10

some additional summations, which is much less computationthan previous FFT-based methods. In fact, using the proposedmethods, we can predict the variance map in much less timethan it takes to reconstruct a single image. The proposedapproximations are especially useful in the case that thevariance information is needed for many or all pixels, suchas when choosing space-varying regularization parameters[6]. The empirical results from the simulated fan-beam CTtransmission scans demonstrate that the proposed varianceapproximations are very accurate. Future work will exploreusing these predictions for regularization design.

Although we focused on variance prediction, by using (15)we could also easily predict covariances and thus predict thecovariance of an ROI whose size is small enough that the localapproximations are sufficiently accurate. However, if only asingle local autocorrelation function is needed, then the FFTapproach is probably easier to use. For analysis of detectabilityof lesions with unknown locations, autocorrelations at manyspatial positions may be needed [10], [25]–[28], in which casethe proposed approach based on (15) can save computation.The matrix method described in (6) and (7) is also applicableto other imaging modalities, such as PET and SPECT [1].Therefore the proposed methods are also readily extended tothose imaging modalities, with different considerations of theweighting function.

The proposed analytical variance approximations are onlyinvestigated in the case of the shift-invariant detector blur. Wecan also generalize the analysis to shift-variant detector blurwhere the local shift-invariance approximation is applicable,e.g., for varifocal collimators in SPECT. In this case, b(s− s′)is replaced by b(s, s′) in (24) and b0(r, ϕ) in (52) becomes

b0(r, ϕ) , m0(ϕ) b(s0(ϕ) + m0(ϕ) r, s0(ϕ)),

where

s0(ϕ) , Dsd arcsinr0(ϕ)

Ds0.

This paper has focused on 2D fan-beam geometry. 3Dgeneralization of these methods can be done by applyingthe same principles [29]. This paper has also focused onanalytical variance approximations for the case of quadraticregularization. An interesting challenge for future work is togeneralize the analysis to the case of edge-preserving non-quadratic regularization. The analysis in [30] may be a usefulstarting point.

APPENDIX IDERIVATION OF LOCAL IMPULSE RESPONSE

Reparameterize variables s and β in (27) according to theinversion of (20) and (21):

s→ s(r) = Dsd arcsin(r/Ds0)

β → β(r, ϕ) = ϕ− arcsin(r/Ds0) .

Then the fan-to-parallel beam rebinning of a(s, β;~n) is

a(s(r), β(r, ϕ);~n) ≈∫

b(s(r)− s(r′))∆

· g(

r′ − rϕ[~n]

∆, ϕ

)

|s(r′)| dr′,

, a(r, ϕ;~n) (46)because r(s(r′)) = r′ and ϕ(s(r′), β(r, ϕ)) ≈ ϕ for r ≈ r′.

A first-order Taylor expansion of s(r) around r′ yieldss(r)− s(r′) ≈ s(r′)(r − r′).

Substituting into (46), the system matrix elements become

a(r, ϕ;~n) ≈∫

b(s(r′)(r − r′)) ∆

· g(

r′ − rϕ[~n]

∆, ϕ

)

|s(r′)| dr′ . (47)

Substituting (47) into (27) and changing variables from (s,β) to (r, ϕ) using (20) and (21) yields the local impulseapproximation,

hd[~n;~n′] ≈ 1

∆β

1

∆S

∫ 2π

0

∫ ∞

−∞

w(r, ϕ)a(r, ϕ;~n)a(r, ϕ;~n′)

· |J(r)| dr dϕ

=1

∆β

1

∆S

∫ 2π

0

w(ϕ;~n;~n′)hϕ[~n;~n′] dϕ, (48)

where |J(r)| = |s(r)| is the determinant of Jacobian matrix,and

hϕ[~n;~n′] ,

∫ ∞

−∞

a(r − rϕ[~n], ϕ;~n) a(r − rϕ[~n′], ϕ;~n′) dr (49)

w(ϕ;~n;~n′) ,

∫∞

−∞w(r, ϕ) |J(r)| a(r − rϕ[~n], ϕ) a(r − rϕ[~n′], ϕ) dr∫∞

−∞a(r − rϕ[~n], ϕ) a(r − rϕ[~n′], ϕ) dr

w(r, ϕ) , w(s(r), β(r, ϕ)).

Let r0(ϕ) , rϕ[~n0]. Because s(r) is fairly smooth, we makethe following approximation for r′ ≈ r0(ϕ):

s(r′) ≈ s(r0(ϕ)) , m0(ϕ) . (50)Substituting (50) into (47) and simplifying yields

a(r, ϕ;~n) ≈ a(r, ϕ;~n0) , a0(r, ϕ)

=

∫

b0(r − rϕ[~n]−r′′, ϕ)∆g

(

r′′

∆, ϕ

)

dr′′, (51)

with r′′ = r′ − rϕ[~n] andb0(r, ϕ) , m0(ϕ) b(m0(ϕ) r) . (52)

Therefore, we further simply (48) as follows

hd[~n;~n′] ≈ 1

∆β

1

∆S

∫ 2π

0

w0(ϕ)h0(∆(~n− ~n′) · ~eϕ, ϕ) dϕ, (53)

where because w(r, ϕ) often varies slowly in r relative to thetypically sharp peak of a0(r, ϕ) at r = 0,

h0(r, ϕ) , a0(r, ϕ) ? a0(r, ϕ),

w(ϕ;~n;~n′) ≈∫∞

−∞w(r, ϕ) |m0(ϕ)| a2

0(r − rϕ[~n0], ϕ) dr∫∞

−∞a20(r − rϕ[~n0], ϕ) dr

≈ |m0(ϕ)| w(r0(ϕ), ϕ) , w0(ϕ), (54)where ? denotes a 1D autocorrelation with respect to r.

11

APPENDIX IIDERIVATION OF LOCAL FREQUENCY RESPONSE

The simplest approach to finding the local frequency re-sponse would be to take the 2D Fourier transform of thelocal impulse response in (28), while ignoring the “edgeeffects” caused by the support functions in (25). We found thisapproach to yield suboptimal accuracy. One way to considerthe edge effects is to use a triangular function with the angular-dependent width:

hϕ[~n;~n0] , h0(∆~n · ~eϕ, ϕ)η1(∆~n)

where

η1(~x) , tri(

~x · ~e⊥ϕd0(ϕ)

)

, (55)

and d0(ϕ) denotes the length of the chord through ~n0 throughthe FOV at angle (ϕ + π/2). This approach is inspired bycirculant approximations for Toeplitz matrices [13], [31], [32],preserving the nonnegative definite property of A′WA. Thischoice might not be optimal in our application, and furtherinvestigation may be beneficial.

One alternative way to consider edge effects is to use thecoordinate transformation recommended for analyzing quasi-stationary noise in [12, p. 870] as follows:

hϕ[~n;~n0] , hϕ[~n0 + ~n/2;~n0 − ~n/2]

= h0(∆~n · ~eϕ, ϕ)η2(∆~n),

where ~x0 , ~xc[~n0], and η2 denotes the support of the image,

η2(~x) , η(~x0 + ~x/2)η(~x0 − ~x/2). (56)

This approach yields a local impulse response that is symmet-ric in ~n, thus ensuring that its spectrum is real.

Another alternative is to refer all displacements relative tothe point ~n0 as follows:

hϕ[~n;~n0] , hϕ[~n0 + ~n;~n0]

= h0(∆~n · ~eϕ, ϕ)η3(∆~n),

whereη3(~x) , η(~x0 + ~x)η(~x0). (57)

This choice is not symmetric in ~n but it better corresponds tothe local Fourier analysis based on the DFT of A′WAej . Forsimplicity, we could also approximate (57) as follows:

η4(~x) ≈ η0(~x) = η(~x)η(~x0). (58)

This choice also yields a local impulse response that issymmetric in ~n provided η(~x) is symmetric itself.

We focus on η1(~x) in (55) hereafter because it preserves theproperty of nonnegative definiteness [33]. Define the following“strip like” function:

sϕ(~x) , h0(~x · ~eϕ, ϕ)η1(~x),

and sϕ(~x)2D FT←→ Sϕ(u, v) . Taking the DSFT of (53) yields

the following result:

Hd0(~ω) =1

∆β

1

∆S

∫ 2π

0

w0(ϕ)Hϕ(~ω) dϕ, (59)

where Hϕ(~ω) is the spectrum of hϕ[~n;~n0] = sϕ(∆~n), asfollows:

Hϕ(~ω) =∑

~n

sϕ(∆~n) e−ı(~ω·~n)

≈ 1

∆2

∫∫

sϕ(~x) e−ı 1

∆(~ω·~x) d~x

=1

∆2Sϕ

(

~ω

2π∆

)

. (60)

The 2D FT of sϕ(~x) is easiest to see for the case ϕ = 0:

s0(x, y) = h0(x, 0)tri(

y

d0(0)

)

2D FT←→ S0(u, v) = |A0(u, 0)|2 d0(0) sinc2(d0(0) v),

where A0(ν, ϕ) is associated with the detector response andbasis effect, given in (44). By the rotation property of the2D FT:

Sϕ(ρ,Φ) ≈ |A0(ρ cos(Φ− ϕ), ϕ)|2

·d0(ϕ) sinc2(d0(ϕ)ρ sin(Φ− ϕ)) .

Therefore, using (54) and (60), the local frequency responseH0(ρ,Φ) around a point ~n0 is

H0(ρ,Φ) ≈ 1

∆β∆S∆2

∫ 2π

0

w0(ϕ)Sϕ(ρ,Φ) dϕ . (61)

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